To do part two you need to use the "standard" Young's tableaux, which are obtained by filling in the frames with integers in standard order, left to right and top to bottom. Use primed integers for the second factor. Thus the standard frames for (2,1) x (2,1) are:

1 2 3
4

1' 2' 3'
4'

You decomposed the direct product by starting with the first factor and attaching the squares of the second factor to it in all possible ways. So now in the tableau, consider the permutation 1 ↔ 1', 2 ↔ 2', etc. This is an involution, and therefore either an odd or even permutation. Even means that rep belongs to the symmetric part of the product, odd means it's antisymmetric.

For example, take [42], which was

xxxxx
xxx

The standard tableau is

1 2 3 1' 2'
4 3' 4'

Switching the 1's, 2's and 4's are all even since they only involve horizontal moves. But switching 3 with 3' is a vertical move, hence odd. We conclude therefore that [42] is antisymmetric.